Thomas Carraro

Indirect Multiple Shooting for Nonlinear Parabolic Optimal Control Problems with Control Constraints

We discuss the indirect multiple shooting approach for the solution of PDE-based parabolic optimal control problems with control constraints. The method is formulated within an abstract function space setting and uses a space-time Galerkin finite element discretization. The emphasis is on the embedding of indirect multiple shooting into the optimal control framework as well as the detailed description of the discretization within the PDE context. Numerical results for linear and nonlinear model problems with and without control constraints illustrate the efficient use of indirect multiple shooting particularly in cases where other standard methods fail.

Direct and Indirect Multiple Shooting for Parabolic Optimal Control Problems

We present two multiple shooting approaches for optimal control problems (OCP) governed by parabolic partial differential equations (PDE). In the context of ordinary differential equations, shooting techniques have become a state-of-the-art solver component, whereas their application in the PDE case is still in an early phase of development. We derive both direct (DMS) and indirect (IMS) multiple shooting for PDE optimal control from the same extended problem formulation. This approach shows that they are algebraically equivalent on an abstract function space level. However, discussing their respective algorithmic realizations, we underline differences between DMS and IMS. In the numerical examples, we cover both linear and nonlinear parabolic side conditions.

Determination of Kinetic Parameters in Laminar Flow Reactors. I. Theoretical Aspects

This article describes the development of a numerical tool for the simulation, the estimation of parameters and the systematic experimental design optimization of chemical flow reactors. The goal is the reliable determination of unknown kinetic parameters of elementary reactions from measurements in a wide range of (laminar) flow conditions, from low to high temperature and from low to high pressure. The corresponding experiments have been set-up in the physical-chemistry group of J. Wolfrum at the PCI, Heidelberg; see the article Hanf/Volpp/Wolfrum [24] in this volume. The underlying mathematical model is the full set of the compressible Navier-Stokes equations accompanied by the balance equations for the chemical species. This system is discretized by a finite element method with mesh adaptivity driven by duality-based a posteriori error estimates (‘DWR method’); see the article Becker et al. [12] in this volume. The parameter estimation uses the Lagrangian formalism by which the problem is reformulated as a nonlinear saddle-point boundary value problem which is solved on the discrete level by the Newton or Gaus-Newton method. The contents of this article is as follows: Introduction Mathematical model Numerical approach The low-temperature flow reactor The high-temperature flow reactor A step towards optimal experimental design Conclusion and outlook References Appendix

A Goal-Oriented Error Estimator for a Class of Homogenization Problems

We present a goal-oriented a posteriori error estimator for finite element approximations of a class of homogenization problems. As a rule, homogenization problems are defined through the coupling of a macroscopic solution and the solution of auxiliary problems. In this work we assume that the homogenized problem is known and that it depends on a finite number of auxiliary problems. The accuracy in the goal functional depends therefore on the discretization error of the macroscopic and the auxiliary solutions. We show that it is possible to compute the error contributions of all solution components separately and use this information to balance the different discretization errors. Additionally, we steer a local mesh refinement for both the macroscopic problem and the auxiliary problems. The high efficiency of this approach is shown by numerical examples. These include the upscaling of a periodic diffusion tensor, the case of a Stokes flow over a porous bed, and the homogenization of a fuel cell model which includes the flow in a gas channel over a porous substrate coupled with a multispecies nonlinear transport equation.

Coupling vs decoupling approaches for PDE/ODE systems modeling intercellular signaling

We consider PDE/ODE systems for the simulation of intercellular signaling in multicellular environments. The intracellular processes for each cell described here by ODEs determine the long-time dynamics, but the PDE part dominates the solving effort. Thus, it is not clear if commonly used decoupling methods can outperform a coupling approach. Based on a sensitivity analysis, we present a systematic comparison between coupling and decoupling approaches for this class of problems and show numerical results. For biologically relevant configurations of the model, our quantitative study shows that a coupling approach performs much better than a decoupling one.

An Adaptive Newton Algorithm for Optimal Control Problems with Application to Optimal Electrode Design

In this work, we present an adaptive Newton-type method to solve nonlinear constrained optimization problems, in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive strategy is based on a goal-oriented a posteriori error estimation for the discretization and for the iteration error. The iteration error stems from an inexact solution of the nonlinear system of first-order optimality conditions by the Newton-type method. This strategy allows one to balance the two errors and to derive effective stopping criteria for the Newton iterations. The algorithm proceeds with the search of the optimal point on coarse grids, which are refined only if the discretization error becomes dominant. Using computable error indicators, the mesh is refined locally leading to a highly efficient solution process. The performance of the algorithm is shown with several examples and in particular with an application in the neurosciences: the optimal electrode design for the study of neuronal networks.

Successive Approximation of Nonlinear Confidence Regions (SANCR)

In parameter estimation problems an important issue is the approximation of the confidence region of the estimated parameters. Especially for models based on differential equations, the needed computational costs require particular attention. For this reason, in many cases only linearized confidence regions are used. However, despite the low computational cost of the linearized confidence regions, their accuracy is often limited. To combine high accuracy and low computational costs, we have developed a method that uses only successive linearizations in the vicinity of an estimator. To accelerate the process, a principal axis decomposition of the covariance matrix of the parameters is employed. A numerical example illustrates the method.

Parameter Estimation for a Reconstructed SOFC Mixed-Conducting LSCF-Cathode

The performance of a solid oxide fuel cell (SOFC) is strongly affected by electrode polarization losses, which are related to the composition and the microstructure of the porous materials. A model that can decouple the effects associated with the geometrical arrangement, shape, and size of the particles together with material distribution on one side and the material properties on the other can give a relevant improvement in the understanding of the underlying processes. A porous mixed ionic-electronic conducting (MIEC) cathode was reconstructed by focused ion beam tomography. The detailed geometry of the microstructure is used for 3D calculations of the electrochemical processes in the electrode and to calibrate a well-established reduced model obtained by averaging. We perform a model-based estimation of the parameters describing the main processes and estimate their confidence regions using the calibrated reduced model.